You can easily use adsense with blogger. There is a blogger adsense gadget, but I recommend you create the adsense unit through your adsense account and insert it into your blog via the HTML gadget.
The adsense widget is limited, so using the manual approach provides you with more freedom such as customizing your adsense ads and using channels to track your adsense data.
Thaks for the replys! I know you can use blogger with your website, but what if I use the free blogger (ex: yourblog.blogspot.com). How would you get traffic with that .blogspot.com at the end of your name?
You can use domain names with Blogger too. Although, people have been successful with blogspot in their name, but you should look into registering a domain name.
Blogger has information in the help about using custom domain names.
It may help traffic. For me the largest benefit of buying a domain name was when I switched to Wordpress. Since I had my domain name with blogger, the address of my blog hadn't changed with the move. This meant that any backlinks that I had acquired still worked after the move.
As for the Monetize button, I would manually add adsense into your blog instead of using the monetize button. Wil the button makes things easy, you lose such things as customizing your ads and channels.